In this article, we analytically investigate rogue wave patterns in the NLS equation at arbitrary rogue-wave orders. We show that if any internal parameter in the rogue waves (which controls the shape of initial small perturbations to the uniform background) is large, then these waves would exhibit clear geometric structures, which comprise Peregrine (fundamental) rogue waves forming shapes such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at the center. These rogue patterns are analytically predicted by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controlled by the phase of the large internal parameter. We also show that if multiple internal parameters in the rogue waves are large but satisfy certain constraints, then the same rogue patterns would still hold. These results reveal a deep connection between rogue patterns and the Yablonskii-Vorob'ev polynomials, and drastically improve our analytical understanding and quantitative description of rogue events. As a small application of our analytical results, the numerical observation in [1]} on single-shell ring structures is explained. Comparison between true rogue patterns and our analytical predictions is also presented, and excellent agreement is observed.
